Optimal. Leaf size=85 \[ -\frac {6 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {6 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
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Rubi [A] time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3362, 3296, 2637} \[ -\frac {6 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {6 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3362
Rubi steps
\begin {align*} \int \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {3 \operatorname {Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {6 \operatorname {Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d}\\ &=\frac {6 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {6 \operatorname {Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d}\\ &=\frac {6 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {6 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {3 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 65, normalized size = 0.76 \[ \frac {3 \left (b^2 (c+d x)^{2/3}-2\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )+6 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 57, normalized size = 0.67 \[ \frac {3 \, {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} b \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 81, normalized size = 0.95 \[ \frac {3 \, {\left (\frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b} + \frac {{\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + a^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{2}}\right )}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 131, normalized size = 1.54 \[ \frac {3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 a \left (\cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )+3 a^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{d \,b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 118, normalized size = 1.39 \[ \frac {3 \, {\left (a^{2} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a + 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 68, normalized size = 0.80 \[ \frac {6\,b\,\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{1/3}-6\,\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )+3\,b^2\,\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{2/3}}{b^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.16, size = 94, normalized size = 1.11 \[ \begin {cases} x \cos {\relax (a )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \cos {\left (a + b \sqrt [3]{c} \right )} & \text {for}\: d = 0 \\\frac {3 \left (c + d x\right )^{\frac {2}{3}} \sin {\left (a + b \sqrt [3]{c + d x} \right )}}{b d} + \frac {6 \sqrt [3]{c + d x} \cos {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{2} d} - \frac {6 \sin {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{3} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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